Allen Shields*

Felix Hausdorff: Grundziige der Mengenlehre

The first edition of Hausdorff's classic Principles of Set Theory appeared 75 years ago. Although there had been other treatments of the material somewhat earlier (see the Anhang, p. 449 for references), this is the book from which succeeding generations of mathema- ticians learned the elements of set theory and point set topology. The Introduction begins with the statement that the book is intended to be a textbook and not a monograph, and that complete proofs are given. Hausdorff states that W. Blaschke (Prague) drew all the figures.

The first two chapters deal with sets and functions. The notations are not modern: If A and B are sets than their union and intersection are denoted (~(A,B) and ~(A,B). If the sets are disjoint, then the union is also denoted by A + B. The difference of two sets, denoted B - A, is defined only when A is a subset of B. On page 14 "rings" and "fields" of sets are introduced: a family of sets is called a ring if it is closed under finite unions and intersections, a family of sets is called a field if it is closed under differences and finite unions. A footnote states that the terminology is taken from the theory of algebraic numbers but that there is only a partial analogy which should not be pushed too far. A family of sets is called a c-family if it is dosed under countable unions; it is called a g-family if it is closed under countable intersections. Thus one has or-rings and or-fields.

The third chapter deals with cardinal numbers and their arithmetic, including exponents. Here F. Bern- stein's theorem is proved: if A and B are sets, if there is a one-to-one map of A onto a subset of B, and if there

* Column editor's address: Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1003 USA

is also a one-to-one map of B onto a subset of A, then there is a one-to-one map of A onto B. Corollary: If a and b are cardinal numbers and if both a ~ b and also b ~ a, then a = b. Note, however, that at this point one does not yet know that any two cardinal numbers are comparable; this is proved in Chapter 5. Finally in this chapter it is shown that if c denotes the cardinal number of the continuum, then c c = 2 c.

The fourth chapter deals with linearly ordered sets, the fifth with well-ordered sets, and the sixth with re- lations between ordered and well-ordered sets. Or- dinal numbers and transfinite induction are intro- duced in Chapter 5, as is the first transfinite number 00. This chapter also contains Zermelo's proof that every set can be well ordered. The Axiom of Choice appears in the following form: if M is a nonempty set then to each nonempty subset A of M there is asso- ciated a distinguished element a = f(A) of A. Appar- ently Hausdor f f regards the existence of such a "choice function" f as obvious; at any rate the exis- tence is merely asserted (p. 136) with no further dis- cussion. The cardinal numbers associated with well ordered sets are called alephs; because of the well-or- dering theorem every cardinal number is an aleph. It now follows that any two cardinals are comparable. Theorem: Every infinite aleph is equal to its square. The proof uses transfinite induction and is taken from Jourdain [1908]; the first (complete) proof is in Hes- senberg [1906] p. 593.

Chapter 6 (p. 141) contains the following result, sometimes called Hausdorff's Maximal Principle: If A is a partially ordered set, then A contains maximal linearly ordered subsets. This is proved using the well-or- dering theorem of Zermelo. This is of course equiva- lent to Zorn's Lemma, which was only published 21 years later (see Zorn [1935]). J. L. Kelley in the ap- pendix to Kelley [1957] emphasizes the Hausdorff

6 THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1 �9 1989 Springer-Verlag New York

Maximal Principle and uses it throughout the book. Hausdorff himself does not make as much use of it, often preferring to use transfinite induction. Nor does Hausdorff show that his principle implies the well-or- dering theorem or the Axiom of Choice.

Much of the material in Chapters 4, 5, and 6 is quite specialized and not of current interest.

The last four chapters, 7, 8, 9, and 10, introduce to- pological ideas and related concepts. After defining metric spaces in Chapter 7, the author defines what are today called Hausdorff spaces (p. 213). This seems to be the place where abstract topological spaces were first defined. (Compare Weyl [1913], w where ab- stract two-dimensional Hausdorff manifolds are de- fined in terms of Euclidean neighborhoods satisfying certain axioms.) Hausdorff states in the notes at the end of the book (pp. 456-457) that he first developed this theory (Hausdorff spaces) in lectures at the Uni- versity of Bonn in the summer semester of 1912. He goes on to discuss open sets, closed sets, limit points, boundary points, etc. Following Fr6chet a subset E of a topological space is said to be compact if every infinite subset of E has a limit point (which need not belong to E). Thus compact sets in this sense need not be closed. Hausdorff then proves (Borel's theorem) that every countable open cover of a closed compact set has a fi- nite subcover. He also proves a partial converse: if every open cover (not necessarily countable) of E con- tains a finite subcover, then E is dosed and compact. In the notes (p. 457) he states: "'Many proofs of Borel's theorem have been given, some of them unimaginably complicated, compare Schoenflies [1913] p. 234." Fi- nally, in w of Chapter 7 (p. 244) Hausdorff gives the modem definition of connected space. He states in a footnote that many other definitions have been given, all different from that of the text, which seems to the author to be the most natural and the most general.

Chapter 8, entitled "Point sets in special spaces," is by far the longest chapter in the book, almost 100 pages long. In w p. 263, the first and second axioms of countability (namely, each point has a countable neighborhood basis; and, the whole space has a coun- table basis) are introduced and discussed. In w a number of examples are gathered together with ap- propriate metrics: the space of all sequences (intro- duced by Fr6chet), (~2, C(0,1) with the supremum norm, and C(0,1) with the L 2 norm. In w the non- empty closed subsets of a closed compact set of a metric space are made into a metric space, with what is nowadays called the Hausdorff metric. The defini- tion of this metric is equivalent to the following: d(A, B) = max{AB, BA}, where AB denotes the in- fimum of ~ such that B C A,, and A, denotes the set of points at distance less than e from A. Hausdorff states that the quantity AB occurs in Pomp6iu [1905].

In w of Chapter 8 Hausdorff shows (using a result of Liouville on the approx imat ion of algebraic

numbers by rational numbers) how to construct a Gs set that contains all the rational numbers, but does not contain any irrational algebraic numbers. [Note by the column editor: LiouviUe's theorem is unnecessary here. It is easy to show directly that if D is any denu- merable set of real numbers disjoint from the ra- tionals, then there is a G8 set containing all the ra- tionals but not containing any point of D.] In w the concept of a totally bounded subset of a metric space is introduced, and its relation to compactness is dis- cussed. w discusses complete metric spaces. On p. 319 W. H. Young's theorem is proved: In a complete metric space an uncountable G~ set has at least the car- dinality of the continuum (see Young [ 1906], Theorem 31, p. 64). (Earlier Cantor had proved this for dosed subsets of the real line.) This was a first step toward proving the corresponding result for general Borel sets, i.e., for verifying the continuum hypothesis for Borel sets; this was achieved two years later indepen- dently by Aleksandrov [1916] and by Hausdorff [1916]. w studies Euclidean spaces, while w studies topo- logical properties of the Euclidean plane; in particular, the Jordan curve theorem is proved. The proof is taken from Brouwer [1910], but Hausdorff states that the method is very close to that used by Veblen [1905]. [Query by the column editor: Who gave the first com- plete proof of the Jordan curve theorem? Veblen's proof was from his dissertation, written at the Univer- sity of Chicago under E. H. Moore.]

Chapter 9 studies functions in topological spaces. Theorem II of w is the "Invariance of domain" for planar sets: If A and B are subsets of the Euclidean plane, with A an open set, and if B is a one-to-one continuous image of A, then B is an open set, and the inverse map from B to A is continuous. This is due to J(irgens [1879]. (The n-dimensional analogue is due to Brouwer.) The remaining sections of this chapter study functions and sequences of functions.

The tenth and last chapter studies measure and con- tent. The idea is to assign a nonnegative number f(A), the measure or content of A, to each set A in some collection of sets. The author states that the theory of measure developed in two stages: the first stage began with G. Cantor and H. Hankel, and was completed by G. Peano and C. Jordan. The principal requirement was finite additivity: f(A U B) = f(A) + fiB) when A and B are disjoint. The second stage, developed by E. Borel and H. Lebesgue, required the stronger condition of countable additivity. Hausdorff comments (p. 400): "'The transition from finite to countable additivity in the new measure and integration theory must be re- garded as one of the greatest advances in mathe- matics."

In the notes (p. 451) Hausdorff considers a ring ~ of sets, with a nonnegative function fdef ined on ~ , such that f vanishes at the null set, and if X, Y E 9d?, with A = X U Y, B = X N Y, then fiX) + flY) = f(A) + fiB).

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989 7

He proves that f extends uniquely to the smallest field of sets containing ~d~.

By far the most famous part of this chapter occurs in the notes (p. 469). Here Hausdorf f presents his famous "paradox." Two subsets of the 2-sphere will be said to be congruent if there is a rotation of the sphere carrying one set onto the other. Hausdorff de- composes the sphere into four subsets, A, B, C, D, such that D is a countable set, A, B, and C are pairwise congruent, and finally, A is congruent to B U C! Thus there does not exist a finitely additive measure defined

This is the book f r o m wh ich succeeding gener- a t ion o f m a t h e m a t i c i a n s learned the e lements o f se t theory and p o i n t se t topology.

on all subsets of the 2-sphere, finite valued and not identically zero, such that congruent sets have the same measure. (Denumerable sets automatically have measure zero. Indeed, if n is a positive integer then one can choose n rotations of the sphere S such that the n images of D are all disjoint. Since these sets all have the same measure we have: nf(D) ~ f(S).) Haus- dorff's proof depends on finding two rotations ~ and qJ such that ~2 = 1, & = 1, and there are no other rela- tions between q~ and ~.

Such measures do exist on the circle group (see, for example, Banach [1932], Chap. II, w However even on the circle such a measure cannot be countably additive. Indeed, on pp. 401-402 Hausdorff decom- poses the circle into countably many pairwise con- gruent sets.

The Hausdorff paradox was generalized in a famous paper by Banach and Tarski [1924]. For recent surveys see Wagon [1985] and French [1988].

Later in Chapter 10 Hausdorff develops the theory of Peano-Jordan content, then Lebesgue measure (on the line) and the Lebesgue integral (the integral of a positive function is defined as the measure of the set lying under the curve). Some of the convergence theorems are presented, as well as the basic differen- tiation theory.

Thus Hausdorff's book served as an introduction to set theory, point set topology, as well as real analysis.

We conclude with some information about Haus- dorff's life, taken from a short article by W. Purkert (see Beckert-Purkert [1987], pp. 201-204). Additional information, including his publications under the name Paul Mongr6 (see below) may be found in DMV [1967].

Felix Hausdorff was born in Breslau on 8 November 1868, the son of a well-to-do merchant; the family

moved to Leipzig in 1871. He graduated (promovierte) from Leipzig University in 1891 in astronomy, under H. Bruns. He completed the Habilitation in 1895 with a work entitled: "On the absorbtion of light in the at- mosphere," and became a Privatdozent, supported by his father. (Dozents were not paid a salary, they re- ceived only the small fees paid by students who at- tended the lectures.) During this period Hausdorff also wrote poems, and at least one successful play, under the pseudonym Paul Mongr6. In 1899 he mar- ried Charlotte Goldschmidt; they had one daughter.

About 1900 Hausdorff became interested in Cantor's set theory. He lectured on it to three students in the summer semester of 1901. This may have been the first lecture course on set theory anywhere in Germany; Cantor himself, in his more than 40 years at Halle, never lectured on set theory. In a letter to Hilbert in 1907 Cantor indicated that he had suggested to Haus- dorff some problems on ordered sets, and that he was pleased with Hausdorff's results.

In 1901 Hausdorff was proposed for an associate (ausserordentliche) professorship at Leipzig. The faculty vote was 22 in favor and seven opposed. Before sending this on to the Minister, who had the final power to make the appointment, the Dean added a note stating that the minority had voted against the appointment because Hausdorff was of the "faith of Moses." He did receive the appointment.

In 1910 Hausdorff went to Bonn as an associate pro- fessor, then in 1913 to Greifswald as a professor, and finally he returned to Bonn in 1921 as a professor. In 1935 the Nazis compelled him to retire. He was still permitted to publish until 1938. After that he prepared several manuscripts which went into storage; some were published after the War (see Hausdorff [1969]). On 26 January 1942, with deportation threatening, Hausdorff, his wife, and her sister committed suicide.

Bibliography

Abbreviations: JFM = Jahrbuch ~iber die Fortschritte der Mathe- matik, MR = Mathematical Reviews, ZBL = Zentralblatt far Mathematik.

P. S. Aleksandrov [1916], Sur la puissance des ensembles mesurables B, Compt. rend. Acad. Sci. Paris 162, 323-325. JFM 46, 301.

S. Banach [1932], Th6orie des op6rations lin6aires, Warsaw. JFM 58, 420; ZBL 5, 209.

S. Banach and A. Tarski [1924], Sur la d6composition des ensembles de points en parties respectivement con- gruentes, Fund. Math. 6, 244-277. JFM 50, 370-371.

H. Beckert, W. Purkert [1987], Leipziger mathematische An- trittsvorlesungen, edited by H. Beckert and W. Purkert, Teubner Archiv zur Mathematik, Band 8, B. G. Teubner Verlagsgesellschaft, Leipzig.

L. E. J. Brouwer [1910], Beweis des Jordanschen Kurven- satzes, Math. Ann. 69, 169-175. JFM 41, 544.

DMV [1967], Felix Hausdorff zum Ged/ichtnis, Jahresb. Deutsch. Math. Verein. 69, 51-76. MR 34 #7330.

8 THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

R. M. French [1988], The Banach-Tarski Theorem, Mathemat- ical Intelligencer, 10(1988), No. 4, 21-28.

F. Hausdorff [1914], Grundz~ige der Mengenlehre, Veit & Co., Leipzig. Reprint: Chelsea Publishers, New York 1949. JFM 45, 123.

- - [1916], Die M/ichtigkeit der Borelschen Mengen, Math. Ann. 77, 430-437. JFM 46, 291.

- - [1969], Nachgelassene Schriften, edited by G. Berg- mann, Stuttgart, Teubner-Verlag. MR 39, 5306.

G. Hessenberg [1906], Grundbegriffe der Mengenlehre, Vandenhoeck & Ruprecht, GOttingen. JFM 37, 67-68.

Ph. Jourdain [1908], On the multiplication of alephs, Math. Ann. 65, 506-512, JFM 39, 101.

E. Ji~rgens [1879], Allgemeine S/itze ~iber Systeme von zwei eindeutigen und stetigen reellen Funktionen von zwei reellen Ver/inderlichen, B. G. Teubern, Leipzig. JFM I1, 269-270.

J. L. Kelley [1957], General topology, van Nostrand, Princeton. MR 16, 1136.

D. Pomp6iu [1905], Sur la continuit6 des fonctions de vari- ables complexes, Ann. Fac. Toulouse 7, 264-315. JFM 36, 454.

A. Schoenflies [1913], Entwicklung der Mengenlehre und ihrer Anwendungen, erste H/ilfte, B. G. Teubner, Leipzig and Berlin. JFM 44, 87.

O. Veblen [1905], Theory of plane cur~es in non-metrical analysis situs. Trans. Amer. Math. Soc. 6, 83-98. JFM 36, 530.

S. Wagon [1985], The Banach-Tarski paradox, Encyclop. Math. and Applic., vol. 24, Cambridge University Press. MR 87e:04007.

H. Weyl [1913], Die Idee der Riemannschen F1/iche, B. G. Teubner, Leipzig; [1923] 2 na ed.; [1955] 3 'd ed. (Stutt- gart). MR 16, 1097. English translation: [1964] The con- cept of a Riemann surface, Addison-Wesley, Reading, Mass. MR 29 #3628.

W. H. Young and Grace Chisholm Young [1906], The theory of sets of points, Cambridge University Press. JFM 37, 70.

M. Zorn [1935], A remark on method in transfinite algebra, Bull. Amer. Math. Soc. 41, 667-670, JFM 61, 1028, ZBL 12, 337.

C o m m e n t s o n p a s t c o l u m n s . Vol. 9, No. 2. At the end of the column we quoted from P. S. Aleksandrov's in- troduction to the Russian edition of Kelley [1957] where he refers to Moore-Smith convergence. The usual reference is to Moore-Smith [1922]. Aleksandrov (p. 7) states that "this same concept was discovered two decades earlier by the talented Odessa mathema- tician S. O. ~atunovski." However Aleksandrov does not give a reference. Since then V. I. Arnol 'd has kindly lent me a book, ~atunovski [1923], which partly clarifies the matter. Arnol 'd had obtained the book from his father, I. V. Arnol'd, who was also a mathe- matician and who had attended classes taught by ~a- tunovski. We quote from the introduction.

The appearance in print of this book I owe to my pupil I. V. Arnol'd who took on himself the labor of preparing for print my lectures on Introduction to Analysis, given at the Novorossisk University and then at the Odessa Insti- tute of Popular Instruction [Narodnogo Obrazovaniya]. The primary purpose of the course was to establish the concept of real number.

He goes on to say that he will introduce conver- gence along certain sets (he calls them manifolds), but he will not assume that these sets are linearly ordered. Instead, he only needs to require that to each two ele- ments in the set there is a third element greater than both of them. He states that one can use this concept to prove the existence of the definite integral. In the text itself he does not discuss integrals, though he does point out that the set of all partitions of a closed interval, partially ordered by mesh size, has the above property.

Thus we can conclude that he had the idea of di- rected sets, at least for some years prior to 1923. There

S. O. ~ a t u n o v s k i

is no indication that he applied the concept to general topology. As regards E. H. Moore, it is quite possible that he too had the idea long before 1922. He spent much of his life working Oh a theory of "general anal- ysis," only part of which was ever published (see Moore [1935]). I remember meeting a former student of his some thirty years ago, who was supposed to be working on Moore's manuscripts; Moore died in 1932.

For a brief biography of ~atunovski see Bogolyubov [1983], p. 29.

Vol. 10, No. 2. In the course of discussing the origin of the modern concepts of differentiable manifold and Riemann surface we referred to the reproduction of Felix Klein's handwritten lecture notes. Such repro- ductions were called "Autographien." In this connec- tion we have received an interesting letter from B. G. Teubner Verlag (701 Leipzig, Goldschmidt Str. 28,

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989 9

German Democratic Republic), the original publishers of these handwritten notes. The letter is signed by Dr. Sc. Genschorek (Cheflektor), and by Weiss (Lektor). They write that for his "Riemann Surfaces" notes, Klein himself wrote them out by hand; they were then hand-copied by collaborators for the printing process.

As to whose handwriting we actually have in the published copy, they say that Klein wrote at various times mentioning people who had helped him with different sets of notes. For example concerning his lec- tures in Leipzig 1880/81 on geometric function theory, he writes that in 1892 the notes for the first semester were written in an Autographie by Paul Epstein (ear- lier the lecture notes had been reworked by Ernst Lange). [Note: Epstein is remembered, among other things, for the Epstein zeta function. He was dis- missed from Frankfurt Universi ty in 1935 by the Nazis, and committed suicide in 1938. See Siegel [1966] pp. 470 for more details.]

Another time Klein states that his assistant, Dr. Ernst Hellinger, "excellently qualified," prepared the Autographie. [Hellinger later worked in integral equa- tions and operator theory. He was dismissed from Frankfurt University in 1935, emigrated to the USA, and finished his career as a professor at Northwestern University.] Still another time it was Ft. Schilling who prepared the Autographie.

Teubner Verlag has recently (1986) republished Klein's lectures on Riemann Surfaces, with commen- taries by G. Eisenreich and W. Purkert (Teubner-Ar- chiv zur Mathematik, vol.5). They have also published (vol. 7) Klein's Funktionentheorie in geometrischer Be- handlungsweise (the Leipzig lectures, 1880/81), with commentaries by F. K6nig and an introduction by F. Hirzebruch. Two additional books of mathematical historical interest are: Weierstrass's lectures on func- tion theory 1886, and Berlin mathematicians of the nineteenth century. These books are available outside Eastern Europe from Springer-Verlag.

We wish to thank Herren Genschorek and Weiss for this information.

Bibl iography

A. N. Bogolyubov [1983], Mathematicians and Mechani- cians, a biographical reference book, lzdat. Naukova Dumka, Kiev. MR 85f:01003.

J. L. Kelley [1957], General topology, van Nostrand, Princeton. MR 16, 1136. Russian edition: Ob~aya topo- logiya, Izdat. "Nauka", Moscow (1968). MR 39 #907.

E. H. Moore [1935], General analysis, written with the col- laboration of R. W. Barnard, The American Philosoph- ical Society, Philadelphia. ]FM 61, 433.

E. H. Moore and H. L. Smith [1922], A general theory of limits, Amer. J. Math. 44 (1922), 102-121. JFM 48, 1254.

S. O. ~atunovski [1923], Introduction to analysis, Mathess, Odessa.

C. L. Siegel [1966], Gesammelte Abhandlungen III, Springer- Verlag, Berlin and Heidelberg, MR 33, 5441.

10 THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

A Nice Little Earner

Once every decade the UK government produces a survey of the career achievements of university graduates. The latest such exercise has just ap- peared, the National Survey of 1980 Graduates and Diplomates, Department of Employment. (A de- tailed excerpt is given in Nature 334, 4 August 1988, pp. 393-4.) And guess what subject comes out with the highest salaries? For women as well as for men?

Mathematics. Well, in combination with computing, but then

we can all turn our hands to that, can't we? Here is an excerpt of the figures, including a

few non-scientific subjects. It's very instructive, and could scotch a few myths.

Average Salaries in 1986 (s sterling p.a.)

Average Salary

Degree Subject Men Women

Mathematics, Computing 16,610 13,520

Electrical Engineering 15,250 10,480

Physics, Maths/Physics 14,330 11,540

Biochemistry 11,470 10,850

Business Studies, Economics, Law, and Accountancy 16,480 13,420

Architecture, Town Planning, and other vocational subjects 12,650 9,950

We've always known we're like gold dust. Ap- parently, the rest of society agrees!

Ian Stewart